The solution of the differential equation $x dy + y dx - \sqrt{1 - x^2 y^2} dx = 0$ is

The solution of the differential equation $2x \frac{dy}{dx} - y = 0$ with the condition $y(1) = 2$ represents . . . . . . .

The general solution of the differential equation $\frac{dy}{dx} = e^{x-y}$ is . . . . . . .

The solution of the differential equation $y(1+\log x) \frac{dx}{dy} - x \log x = 0$ is

Let a solution $y=y(x)$ of the differential equation $x \sqrt{x^2-1} dy - y \sqrt{y^2-1} dx = 0$ satisfy $y(2) = \frac{2}{\sqrt{3}}$.
$STATEMENT-1$: $y(x) = \sec \left(\sec^{-1} x - \frac{\pi}{6}\right)$
$STATEMENT-2$: $y(x)$ is given by $\frac{1}{y} = \frac{2\sqrt{3}}{x} - \sqrt{1 - \frac{1}{x^2}}$

On solving $\frac{dy}{dx} = \frac{x-y+3}{2x-2y+5}$,the solution obtained is $x = 2(x-y) + \log(t) + c$,find $t$.